The action button could (among other things) mark the row in some way to indicate the action has occurred. The first thing the action should do is check whether that marker already exists; if so, no further action is taken.

You’re going to run into problems, though, if two users push the button at or near the same time. The app only sees the copy of the data on the device. One user won’t see what the other has done until both have synced. It’s only after the sync that one can see what the other has done.